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Components of flip graphs of domino tilings in quadriculated cylinder and torus

Author

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  • Liu, Qianqian
  • Wang, Jingfeng
  • Li, Chunmei
  • Zhang, Heping

Abstract

Let R be a quadriculated surface, possibly with boundary, consisting of unit squares, and each interior vertex being surrounded by 4 squares. A tiling of R is a placement of dominoes (a pair of adjacent squares) so that there are no gaps or overlaps. The flip graph of R is a graph whose vertices are all tilings of R and two tilings are adjacent if we can obtain one from another by a flip (90∘ rotation of a pair of side-by-side dominoes). By using graph-theoretical approach, we prove that the flip graph of 2m×(2n+1) quadriculated cylinder is still connected, but that of 2m×(2n+1) quadriculated torus consists of two isomorphic components. For a tiling t, we associate an integer, called forcing number, as the minimum number of dominoes in t that is contained in no other tilings. As a consequence, we obtain that the forcing numbers of all tilings in 2m×(2n+1) quadriculated cylinder and torus form respectively an integer interval whose maximum value is (n+1)m.

Suggested Citation

  • Liu, Qianqian & Wang, Jingfeng & Li, Chunmei & Zhang, Heping, 2026. "Components of flip graphs of domino tilings in quadriculated cylinder and torus," Applied Mathematics and Computation, Elsevier, vol. 510(C).
    • Handle: RePEc:eee:apmaco:v:510:y:2026:i:c:s0096300325004230
    DOI: 10.1016/j.amc.2025.129697
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